Non-local problems with an integral condition for third-order differential equations

The paper is devoted to the study of the solvability of nonlocal problems with an integral variable $t$ condition for the equations
$$u_{tt}+\left(\alpha\frac{\partial}{\partial t}+\beta\right)\Delta u=f(x,t)$$
($\alpha$, $\beta$ are valid constants, $\Delta$ is Laplace operator by spatial variables). Theorems are proved for the studied problems existence and non-existence, uniqueness and non-uniqueness solutions (having all derivatives generalized by S. L. Sobolev included in the equation).